The original XBox is emulated brilliantly for some games. Apart from one bug and the control differences, I'd find it hard to tell which was HALO 2 or 3 from a short bit of gameplay (short enough for me not to notice differences in textures, plot etc.). The problem was, each game had its own profile for the emulator, as no two games required identical settings, and I assume some static recompilation had to be done as well. They could have it perfect for every game, except it would require to much work to be worth it: some people would buy a 360 because it could still play HALO 1 and 2, but there would be obscure games no one cared about.
Halo: CE and Halo 2 are my only Xbox titles, and I bought both when I ordered my 360. Though I have no idea what you mean AnyOldName, not being able to tell the difference between 2 and 3. I guess I've played both so much that I can tell which one it is and where in the game it is, no matter how short the gameplay is :p
Anyway, wikipedia has a nice compatibility listing for all Xbox games that the 360 emulates. There are a couple of reasons why Xbox emulation on the PC has lagged in comparison to the PS2, GC, and Wii. Blueshogun, the dev of cxbx, made a nice long list, forum search his name, I posted it here a while ago. He said a lot of the issues deal with the sheer complexity of the hardware to emulate, and the fact that the documentation for said hardware is missing, incomplete, or inaccurate. He didn't mention consumer hardware power as a major factor, however.
Also, just wanted to say to the OP that none of us knows the future. No one knows if or when emulators for current gen systems (Wii excluded of course) will be available, or what speed they'll run at when functional. It's all guess-work. Could be decades, or some technological revolution might speed things along. Could be never, out of lack of interest (where's my Tiger Game.com emulator?), technical limitations, or because the world ended (it is 2012 after all). We won't know until people start trying and results start coming in. The only thing most of us can say with utmost certainty is that it ain't gonna run on current hardware. Everything else is a big, fat question mark.
I was meaning that in comparison with HALO CE, HALO 2 looks much better when emulated, and is far closer to HALO 3 in graphical quality.
It may be that HALO 2 has replacement textures in its emulation profile, or it may be that all games except HALO CE upscale well, but, like you, I only have those two games for original XBox, as I got them to complete my collection of HALO games.